Individual Gap Measures from Generalized Zeckendorf Decompositions

Abstract

Zeckendorf's theorem states that every positive integer can be uniquely decomposed as a sum of nonconsecutive Fibonacci numbers. The distribution of the number of summands converges to a Gaussian, and the individual measures on gaps between summands for $m \in [F_n, F_{n+1})$ converge to geometric decay for almost all $m$ as $n\to\infty$. While similar results are known for many other recurrences, previous work focused on proving Gaussianity for the number of summands or the average gap measure. We derive general conditions which are easily checked yield geometric decay in the individual gap measures of generalized Zeckendorf decompositions attached to many linear recurrence relations.